Kepler's Second Law
History: Johannes Kepler (http://worldroots.com/brigitte/gifs/kepler_bild.gif) Johannes Kepler (1571-1630), was a Spanish mathematician and astronomer, who proposed that the Earth and planets move about the sun in elliptical orbits. He presented three fundamental laws of planetary motion as well as doing significant work in optics and geometry. Although Kepler came from a poor background, his intelligence brought him a scholarship to the University of to study for the Lutheran ministry. There, Kepler was introduced to the ideas of Copernicus and developed a passion for them. Kepler went to Prague to assist Danish astrologer, Tycho Brahe. The two did not get along and Brahe worried that Kepler would cast a shadow on him as the debut astronomer of his day. He allowed Kepler to see only part of his capacious data and set him the task of understanding the orbit of the planet Mars. Brahe assumed that this would engage Kepler while Brahe worked on his theory of the Solar system. Ironically, it was specifically the Martian data that permitted Kepler to produce the correct laws of planetary motion, accomplishing a place in the development of astronomy far exceeding that of Brahe. ---- Laws of Planetary Motion After Brahe's death, Kepler acquired his data regardless of Brahe's family's effort to restrain the data from him in the desire of financial profit. It is said that Kepler got hold of the data by illegal measures; it is providential for the development of modern astronomy that he obtained this data. Using Brahe's ample and accurate data allowed Kepler to come upon the realization that the orbits of the planets were ellipses to invent his Three Laws of Planetary Motion. ---- Kepler's Second Law Kepler’s second law of the undisturbed planetary motion: A line joining a planet to its star sweeps out equal areas in equal intervals of time. This law is also known as “Kepler’s law of equal areas”. (http://csep10.phys.utk.edu/astr161/lect/history/kepler.html) The imaginary line from the Sun to the planet sweeps out equal areas in equal times, thus planets move faster when they are closer to the Sun. Therefore a planet carries out elliptical motion with an angular speed that continually changes as it shifts in its orbit. The point where a planet is closest to the sun is called perihelion. The point of greatest severance between the sun and planet is called aphelion. As a result, according to Kepler's second law, the planet is at its greatest speed when it approaches perihelion and slowest speed when it approaches aphelion. ---- Analysis Presume a planet were to take one day to travel from point A to B. In that day, an imaginary line marking the present distance between the planet and the sun, would sweep out an approximate triangular area. This triangular area is the amount of area swept throughout one day; it is the same amount of area swept everyday, despite the planet’s position in its orbit, therefore the amount of area as well as the amount of time between points A to B and points C to D would be equal to one another. While planets travel in their elliptical orbits, the distance between the planets to the sun differ with time. Since an equal area is swept during any period of time and as the distance from a planet to its orbiting star changes, it can be concluded that in order for the area swept to remain constant, a planet’s speed must vary. Hence, a planet moves faster when it is closer to the sun. The reason for this is that the sun’s gravity increases the planet’s speed as it gets closer to the sun, and slows down the planet’s speed on the way back around. ---- Example An analogical situation that exemplifies that of Kepler’s Law is if an object of heavy mass were to be thrown upwards. As the object rises, it slowly loses speed and then when it reaches its final height it moves very slowly, and finally comes down picking up speed again. ---- Practice Problems 48. The diagram below shows the movement of a planet around the Sun. Area 1 equals area 2. Compared to the time the planet takes to move from C'' to ''D, the time it takes to move from A'' to ''B is (1) less (2) greater (3) the same 56. The diagram below represents the motion of a planet around the Sun, S. The time that the planet takes to go from point 1 to point 2 is identical to the time the planet takes to go from point 3 to point 4. Which statement must be true? (1) The two shaded regions of the diagram have equal areas. (2) The centripical acceleration of the planet is constant. (3) The planet moves at a constant speed. (4) The planet moves faster when it is farthest from the Sun. Excerpted from: "Barron's Review Course Series Let's Review: Physics The Physical Setting" by Miriam A. Lazar, Third Edition. Barron's Educational Series, Inc. 2004 Answers to Practice Problems 48. (3) the same 56. (1) The two shaded regions of the diagram have equal areas. Excerpted from: "Barron's Review Course Series Let's Review: Physics The Physical Setting" by Miriam A. Lazar, Third Edition. Barron's Educational Series, Inc. 2004 ---- References http://www.walter-fendt.de/ph11e/keplerlaw2.htm (This site has intricate animation that was very helpful to my understanding of Kepler’s second law.) http://www-spof.gsfc.nasa.gov/stargaze/Skepl2nd.htm (This is one of the sites that had helped me the most. The information on this site is very thorough and in depth.) http://www.windows.ucar.edu/tour/link=/the_universe/uts/kepler2_gif.html (This site helped further enhance my analysis of Kepler’s second law, obtaining written examples that helped me to further comprehend exactly what this law means.) http://csep10.phys.utk.edu/astr161/lect/history/kepler.html (This site gave me the most insight on Johannes Kepler’s scientific history.) http://kepler.nasa.gov/johannes/ (This site has a lot of information of Johannes Kepler’s personal life; the way he'"Barron's Review Course Series Let's Review: Physics The Physical Setting" by Miriam A. Lazar, Third Edition. Barron's Educational Series, Inc. 2004' (This book contains helpful sample problems pertaining to any physic’s topic.) ---- Resources http://curriculum.calstatela.edu/courses/builders/lessons/less/les1/kepler2.html (This site, step-by-step, maps out Kepler’s second law.) http://www.kineticbooks.com/physics/trialpse/13_Gravity%20and%20Orbits/21/sp.html (This site defines Kepler’s second law as well as compares his work to those of other theorists.) http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/circmot/ksl.html (This website provides great animation that nicely illustrates Kepler’s second law.) http://zebu.uoregon.edu/~soper/Orbits/kepler2.html (This site provides colorful and vibrant diagrams.) http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion (This is another “Wikipedia” site that I found to be very informative. It is broad yet covers all three of Kepler’s laws of planetary motion.) http://www-astronomy.mps.ohio-state.edu/~dhw/A161/lecture6.html (This site documents Johannes Kepler's achievements as well as evaluating them.) ----